If x

Question

If  x<x^3<x^2 , then  x^4  must be:

(A) less than x
(B) between  x  and  x^3 
(C) between  x^3  and  x^2 
(D) greater than  x^2 
(E) undefined

sujoykrdatta · Answer

We have: x < x^3 < x^2 Observations: x^2 (even power) is greater than x^3 (a higher power, odd) and x^1 (a lower power, odd) This is possible ONLY if x were a negative number. Again. considering the odd powers: x^3 > x^1 (a higher power is greater than a lower power where x is negative) Thus, x must be a fractional value => -1 < x < 0 Thus, x^4 is a fraction between 0 and 1 (a positive quantity) Looking at the options: (A) less than x => Incorrect since x^4 is positive and x is negative (B) between x and x^3 => Incorrect since x^4 is positive and x and x^3 are negative (C) between x^3 and x^2 => Correct; since x is a negative value between 0 and -1, x^4 must be smaller than x^2; hence is lies between x^3 (negative) and x^2 (a larger positive value) (D) greater than x^2 => Incorrect; since x is a negative value between 0 and -1, x^4 must be smaller than x^2 - higher the power, smaller the value (E) undefined => Incorrect, since x is clearly defined Answer C Alternative: We need to consider 4 regions on the number line and verify x < x^3 < x^2: # x>1: Say, x=2: 2 < 8 < 4 - false # 0 1/16 < -1/2 - Incorrect (B) between x and x^3 => -1/2 < 1/16 < -1/8 - Incorrect (C) between x^3 and x^2 => -1/8 < 1/16 < 1/4 - Correct (D) greater than x^2 => 1/16 > 1/4 - Incorrect (E) undefined => Incorrect, since x is clearly defined Answer C

IMPOSSIBLE7 · Answer

Can we do it like below :-

Multiply x

X^2 < X^4 < X^3

Bunuel

Bunuel · Answer

One might think that this gives the correct answer (C), but x^2 < x^4 < x^3 is NOT correct. By the way, IF x^2 < x^4 < x^3 were correct, what would be the answer? C, x^4 is between x^3 and x^2 ? Or D, x^4 is greater than x^2 ? We should be careful when multiplying an inequality by an unknown: we should keep the sign if we multiply by a positive value and flip the sign when we multiply by a negative value . For example, 2 < 3. If you multiply the inequality by 4 (a positive value) we get: 8 < 12 (keep the sign!) but if you multiply the inequality by -4 (a negative value) we get: -8 > -12 (flip the sign!). So, we cannot multiply x < x^3 < x^2 by x and write x^2 < x^4 < x^3 before considering the sign of x. If x were positive, then yes, we'd keep the sign and get x^2 < x^4 < x^3 BUT if x is negative, then after multiplying, we'd flip the sign and get x^2 > x^4 > x^3. Never multiply (or reduce) an inequality by a variable (or the expression with a variable) if you don't know its sign. If x x^2 > x^3 > x^4 > ... (e.g. \frac{1}{2} > (\frac{1}{2})^2 > (\frac{1}{2})^3 > (\frac{1}{2})^4 > ... ) and we know that's not the case. So, x MUST be negative! Now, we can multiply xx^4>x^3 Answer: C. 9. Inequalities Theory Inequalities Made Easy! Inequalities: Tips and hints Arithmetic with Inequalities Wavy Line Method Application - Complex Algebraic Inequalities Solving Quadratic Inequalities: Graphic Approach Inequalities - Quadratic Inequalities Graphic approach to problems with inequalities Inequalities trick INEQUATIONS(Inequalities) Part 1 INEQUATIONS(Inequalities) Part 2 Questions Inequality and absolute value questions from my collection DS Questions PS Questions For more check Ultimate GMAT Quantitative Megathread Hope it helps.

If x < x^3 < x^2, then x^4 must be:

Prep Toolkit

Top 5 GMAT Debrief Videos

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards

GMAT Problem Solving (PS) Questions